2.2. Electronic band structure

The band structure is modeled in the envelope function approximation, using either the single-band effective mass approximation or a multiband model.

The number of bands is specified in the input file by the keyword NumberOfBands.

Single-band effective mass model can be sufficient for intraband devices where the transitions energies are small compared to the band gap. However, to describe nonparabolicity in intraband devices, 2- or 3-band models are needed. The 3-band modeled is strongly recommended for structures based on electrons in III-V heterostructures.

For interband devices, or devices involving only the valence band, the 8-band model is recommended.

2.2.1. Definition of band offsets

The definition is identical to nextnano++. The database contains the values \(E_\mathrm{c}^\Gamma, E_\mathrm{v,av}, \Delta_\mathrm{so}\) and the band gap parameters \(E_\mathrm{g}^\Gamma(T=0), \alpha, \beta\). The choice UseConductionBandOffset results in different temperature dependence of the heterostructure band offsets via the Varshni formula described in nextnano++.

Note

The band offsets get an additional shift if strain is present.

2.2.2. Single-band model

In this case a 1-dimensional Schrödinger equation is solved:

(2.2.2.1)\[-\frac{\hbar^2}{2m_{\perp}^*(z)} \frac{\partial^2}{\partial z^2} \psi(z) + V(z) \psi(z) = E \psi(z)\]

where \(m_{\perp}^*(z)\) is a position-dependent effective mass along the growth direction.

Effective mass

Effective mass from k.p parameters

If Material{ EffectiveMassFromKpParameters } is set to yes, the effective mass is calculated from the k.p parameters using the following equation:

(2.2.2.2)\[\frac{m_0}{m_{\perp}^*} = S + \frac{E_P(E_g+2\Delta_{\text{SO}}/3)}{E_g(E_g+\Delta_{\text{SO}})}\]

where \(E_P\) is the Kane energy, \(E_g\) the band gap and \(\Delta_{\text{SO}}\) is the spin-orbit splitting [Vurgaftman2001]. The effective mass of the database is ignored in this case.

Effective mass without k.p parameters

On the other hand, if Material{ EffectiveMassFromKpParameters } is set to no, the effective mass is taken directly from the nextnano.NEGF database. Note that the material database can be overwritten in the input file using the Material{ Overwrite{ } } option.

Anisotropic effective mass

In the case of an anisotropic effective masse, the axial \(m_{\perp}^*\) and in-plane \(m_{\parallel}^*\) effective masses can be individually overwritten using ElectronMass and ElectronMassInPlane respectively.

Nonparabolicity within single-band approach (not recommended)

If NonParabolicity is set to yes, the effective mass is evaluated at the energy of the lowest miniband:

\[\frac{m_0}{m^*} = S + \frac{E_P((E_g+E_0)+2\Delta_{\text{SO}}/3)}{(E_g+E_0)((E_g+E_0)+\Delta_{\text{SO}})}\]

where \(E_0\) is the energy difference between the lowest miniband and the conduction band edge.

Note, however, that in this case the effective mass is not state-dependent, i.e. higher energy states have the same effective mass than the lowest miniband. A full nonparabolic description is provided by 2- or 3-band models described below.

2.2.3. 2-band model

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2.2.4. 3-band model

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2.2.5. 8-band model

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2.2.6. Output of effective mass in the multiband case

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2.2.7. Rescaling of \(\mathbf{k} \cdot \mathbf{p}\) parameters (for multiband)

When diagonalizing the \(\mathbf{k} \cdot \mathbf{p}\) Hamiltonian for a given wave vector, if the coefficient \(S(L+1)\) of \(k^4\) in the secular equation is positive, two different \(k\) may correspond to the same eigenenergy. One is the expected correct solution, but the other is an oscillatory solution with a large \(k\), and a smooth wave function may not be obtained. To prevent this, the Material{ RescaleS } option rescales \(S\) to 0 (per default) while maintaining the effective mass of the conduction band.

The effect of rescaling on \(S\) and \(E_P\) is the following:

(2.2.7.1)\[S \to S' E_P \to E_P'\]

while the effective mass at bandedge is conserved.

(2.2.7.2)\[S + \frac{E_P}{E_g} = S' + \frac{E_P'}{E_g}\]

while for 3 bands it corresponds to:

(2.2.7.3)\[S + \frac{E_P(E_g + 2\Delta_\mathrm{SO}/3)}{E_g(E_g + \Delta_\mathrm{SO})} = S' + \frac{E_P'(E_g + 2\Delta_\mathrm{SO}/3)}{E_g(E_g + \Delta_\mathrm{SO})}\]

2.2.8. Smoothing of the \(\mathbf{k} \cdot \mathbf{p}\) parameters (for multiband)

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2.2.9. Oscillator strength

The oscillator strength is calculated from the formula

(2.2.9.1)\[f_{\alpha\beta} = \frac{2|p_{\alpha\beta}|^2}{m_0 (E_\beta - E_\alpha)}\]

Note

The electron mass \(m_0\) entering the above formula is the bare electron mass.

This oscillator strength (sometimes referred to as the unnormalized one) differs from the usual definition in the single band case by the ratio \(m^*/m_0\), i.e. \(\frac{m^*}{m_0}f_{\alpha\beta}\) is called the normalized oscillator strength.

The advantage of this unnormalized definition is that it is general enough to be applied to the multiband case.

Note

In the parabolic single-band model, the usual sum-rule is retrieved by using the normalized definition

(2.2.9.2)\[\sum_{\beta \neq \alpha} \frac{m^*}{m_0} f_{\alpha\beta} = 1\]

Last update: 30/10/2024